University Advanced Quantum Mechanics · topic UAQM-01.09
Time reversal in scattering: reciprocity and detailed balance
Connect the ideas, make your working visible, and test the result. Use the hints when you need a starting point; open the full solution after your attempt.
Further practice
Work through a fresh case before tackling the extension.
Time reversal in scattering: reciprocity, detailed balance: predict a finite change
Reciprocity is the statement that, for a time-reversal-invariant interaction, the amplitude for a scattering process equals, up to a phase, the amplitude for the process with initial and final states swapped and every momentum and spin reversed. Use the supplied topic model Y_out = 0.75*nid*(2*spin+1)*max(tp-287.52,0)*(tp+270.48)/(tp*(tp+1876.54)), where Y_out is RATIO R (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. tp is Proton beam energy Tₚ (lab), measured in MeV. Keep spin (Pion spin hypothesis s_π) = 0; nid (Identical-proton factor f) = 2 fixed. For this exercise use only 353.125 MeV ≤ tp ≤ 746.875 MeV on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. Calculate Y_out at tp = 439.75 MeV and 494.875 MeV. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δtp across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
RATIO R against tp
Predictions from the supplied Time reversal in scattering: reciprocity, detailed balance model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): RATIO R
Horizontal axis (x): tp: Proton beam energy Tₚ (lab) / MeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| tp: Proton beam energy Tₚ (lab) / MeV | RATIO R |
|---|---|
| 353.125 | 0.07794166980400188 |
| 451.5625 | 0.16900145110006481 |
| 550 | 0.24205013618343219 |
| 648.4375 | 0.30384359553722856 |
| 746.875 | 0.35776455464717066 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of tp separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At tp = 439.75 MeV, substitute into Y_out = 0.75*nid*(2*spin+1)*max(tp-287.52,0)*(tp+270.48)/(tp*(tp+1876.54)): Y₀ = 0.159218.
- At tp = 494.875 MeV, the same fixed-parameter model gives Y₁ = 0.202846.
- Signed change: ΔY_out = Y₁ − Y₀ = 0.0436278; the input change is Δtp = 55.125 MeV.
- Average sensitivity is ΔY_out/Δtp = 0.000791433 1 per MeV. Its positive sign means that this response increases over the specified interval.
- This is a secant slope between two settings. It equals a point derivative only for a locally linear relation or in a justified small-interval limit.
Answer Y₀ = 0.159218, Y₁ = 0.202846; ΔY_out = 0.0436278; average sensitivity = 0.000791433 1 per MeV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Extra challenging questions
Work through the linked stages and explain what each result means.
Time reversal in scattering: reciprocity, detailed balance: infer an input with uncertainty
Reciprocity is the statement that, for a time-reversal-invariant interaction, the amplitude for a scattering process equals, up to a phase, the amplitude for the process with initial and final states swapped and every momentum and spin reversed. Use the supplied topic model Y_out = 0.75*nid*(2*spin+1)*max(tp-287.52,0)*(tp+270.48)/(tp*(tp+1876.54)), where Y_out is RATIO R (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. tp is Proton beam energy Tₚ (lab), measured in MeV. Keep spin (Pion spin hypothesis s_π) = 0; nid (Identical-proton factor f) = 2 fixed. For this exercise use only 353.125 MeV ≤ tp ≤ 746.875 MeV on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.27037903, with a bounded reading uncertainty ±0.013108. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown tp by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.039375 MeV, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.
RATIO R against tp
Predictions from the supplied Time reversal in scattering: reciprocity, detailed balance model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): RATIO R
Horizontal axis (x): tp: Proton beam energy Tₚ (lab) / MeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| tp: Proton beam energy Tₚ (lab) / MeV | RATIO R |
|---|---|
| 353.125 | 0.07794166980400188 |
| 451.5625 | 0.16900145110006481 |
| 550 | 0.24205013618343219 |
| 648.4375 | 0.30384359553722856 |
| 746.875 | 0.35776455464717066 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(tp) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(353.125) = 0.0779417 and Y_out(746.875) = 0.357765. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives tp_line = 353.125 + (0.270379 − 0.0779417)(746.875 − 353.125)/(0.357765 − 0.0779417) = 623.911 MeV. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives tp = 593.313 MeV; substituting back gives Y_out = 0.27037903.
- The interpolation error in the input is |tp_line − tp| = 30.5988 MeV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.25727103 and 0.28348703. Sort the roots even if the response decreases: 572.899 MeV ≤ tp ≤ 614.383 MeV.
- The lower and upper input deviations are 20.4139 MeV and 21.0701 MeV. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer tp ≈ 593.313 MeV; compatible interval [572.899, 614.383] MeV. Endpoint interpolation gives 623.911 MeV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.